# Construct Viable Arguments and Critique the Reasoning of Others: How Do You Know? Prove it!

• Addition and subtraction are one-dimensional operations (linear—for example joining two Cuisenaire rods or skip counting on a number line); they are at the same and the lowest level of operations. If both operations appear in the same expression, they are executed in order of appearance, first come first serve ();
• Multiplication and division are two-dimensional operations (as represented by an array or the area of a rectangle), therefore, are at a higher level than addition and subtraction, they must be performed before addition and subtraction and if they both appear in an expression should be treated as first come first serve ();
• If all the four operations: addition, subtraction, multiplication, and division appear in a mathematical expression, the order should be: Two-dimensional operations first and then the one-dimensional operation in order of their appearance ().
• Exponential expressions are multi-dimensional (depending on the size of the exponent, e.g., a 10-cube = 103 is a 3-dimensional expression with an exponent of 3 and a base of 10; therefore, exponentiation operation is more important than multiplication (and division) and definitely higher than addition and subtraction, therefore, must be performed before all of them. Therefore, the order of operations so far is: ();
• Grouping operations are expressions included in groups such as brackets, braces, parentheses either transparent and/or hidden (compound expressions in the numerator and denominator of a fraction, function and radical operations are hidden operations. They may involve some or all of the above operations in multiple forms, therefore, are or higher preference than all of the above operations. In transparent grouping operations, the order is parentheses, braces, and brackets. The hidden grouping operations are performed in the context. Inside a grouping operation, the same order as in the above operations is kept.
• I can count 7 after 9.
• Can you give more efficient strategy?
• I can use blue and black Cuisenaire rods.
• Can you give a strategy without concrete materials?
• I can use Empty Number Line.
• Can you give any of the addition strategies?
• Making ten: (9 + 1 + 6 = 10 + 6 = 16)
• Making ten: (6 + 3 + 7 = 6 + 10 = 16)
• Using doubles: (2 + 7 + 7 = 2 + 14 = 16)
• Using doubles: (9 + 9 – 2 = 18 – 2 = 16)
• Using missing double: ( 8 + 1 + 7 = 8 + 8 = 16)
(b) What strategy can be used to find the difference 17 – 9?
• Using teens number: 17 – 9 = 7 + 10 – 9 = 7 + 1 = 8
• Using making ten: 17 – 9 =10 + 7 – 7 – 2 = 10 – 2 = 8
• Using doubles’ strategy 17 – 9 =18 – 1 – 9 = 18 – 10 = 8
• What to add to 9 to get to 17: 9 + 1 + 7 = 17 = 9 + 8 = 17, so 17 – 9 = 8.
(c) What is the nature of the figure formed by joining the consecutive mid points of a quadrilateral?
• To get a sense of the outcome of this construction, I will first consider a special case of quadrilateral: a square or a rectangle.
• What does the constructed figure look like in such a special case?
• What if the quadrilateral is concave? Is this assumption correct? What is your answer in this case? Why?
• Is it true in both cases?
• Is it true for any quadrilateral?
• Can you prove it by geometrical approach?
• Can you prove it by algebraic approach?
(d) How many prime numbers are even? What is the definition of prime numbers? How many factors does an even number have?
• 2 has 2 factors, namely, 1 and 2
• 4 has 3 factors, namely, 1, 2, and 4.
• 6 has 4 factors, namely, 1, 2, 3, and 6.
• All even numbers, except 2 have more than 3 factors.
What conjecture can you form? For upper grades: Can you predict the nature of any even number? Can you prove that ___ is the only even prime number? Is a square number a prime number? Why is a square number not a prime number? If n is a prime number, what can you say about n + 1? (e) Is the product of two irrational numbers always an irrational number?
• What is the definition of an irrational number?
• Is every number an irrational number?
• Why? Can you prove it?
• If not, why?
Can you give a counter example to justify your answer? (f) Will the range of the data change if every piece of data is increased by 5 points? David says: It will increase by 5. Is he right? Why? Can you prove it? Melanie says: It will not change. Is she right? If not, why? Can you prove it? Can you give a counter example to justify your answer? (g) What other central tendencies are affected by such a change? Why? Explain. One of your classmates just stated: Such a change will not change the median of the data, is this true? Why? When students are given opportunities to make conjectures and build a logical progression of statements to explore the truth of their conjectures, they learn the role of reasoning and constructing arguments. The teacher should constantly ask questions such as: “How did you get it?” “What did you do to get this?” “Can you explain your work?” When teachers ask children to explain their approach to finding solutions and the reasons for selecting the particular approach, children develop the ability to communicate their understanding of concepts and procedures and the ability to trust their thinking. Some questions are applicable to all grade levels:
• What mathematical evidence would support your assumption/ approach/strategy/solution?
• How can we be sure of that ….?
• How could you prove that …?
• Will it work if …?
However, some questions should be at grade level. For example, at the high school level the questions can be more content specific. Question: Your classmate claims that the quadratic equation: 2x2 + x + 5 = 0, has no real solutions. This can be followed by questions such as:
• What is a solution to an equation?
• What is a real solution?
• Do you agree with this claim?
• Why? Why not?
• What information in the equation assures you that it does not have any real solutions?
• How did you determine that this does not have a real solution?
• Can you change the constants in this equation so that it will have two real solutions?
• Only one real solution.
Teachers should analyze general situations by breaking them into special cases and ask students to recognize, use and supply examples, counter examples, and non-examples. This can be exemplified by questions such as:
• What were you considering when …?
• Why isn’t every fraction a rational number?
• Is every rational number a fraction?
• Is every fraction a ratio?
• Is every ratio a fraction?
To help children how to learn to justify their conclusions, communicate them to others, and respond to the arguments of others, teachers can ask questions such as:
• How did you decide to try that strategy?
• Do you agree with David’s statement? “Between two rational numbers, there is always a rational number.”
• Why do you agree?
• How will you find it?
• Why don’t you agree?
• Do you have a counter example?
• Is this statement true for all real numbers?
• Why?
• Is every square a rectangle?
• Why?
Analogies and Metaphors as Aids to Mathematical Reasoning Students’ reading comprehension is improved when their thinking involves the understanding of analogy, metaphor, and simile. Similarly, the use of analogies and metaphors is an example of reasoning in mathematics, particularly in the initial stages of learning a concept. Students need to learn to reason by using analogies and reason inductively about data, making plausible arguments that take into account the context from which the data arose. For that teachers need to follow with questions such as:
• What is different and what is same about this problem and the other you solved before?
• Did you try the method of the previous problem?
• Did it work?
• If it did not work, how did you know it did not work?
• Why did it not work?
• Could it work with some changes in your approach? Why or why not? What changes would you make?
• How did you decide to test whether your approach worked?
One of the important aspects of thinking children need to develop is to know the conditions under which a particular definition, formula, or procedure applies and the parameters of its limitations. To develop this ability, teachers could ask questions about the content under discussion. For example:
• Is 3,468 divisible by 4?
• Yes or no?
• Why? Justify your answer without actually dividing the number by 4.
• Is this number divisible by 12? Why? Justify your answer without actually dividing the number by 4.
• In a fraction , if a = b, and b ≠ 0, then the fraction is equal to 1.
Do you agree wit this statement? If so, can you prove it? If not, can you give or construct a counter example to this situation? Students need focused training and support in comparing the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. For this a teacher may articulate questions that focus on:
• How to differentiate between inefficient and efficient lines of reasoning?
• How to focus and listen to the arguments of others and ask questions to determine if the reasoning and the direction of the argument make sense?
Finally, teachers should ask clarifying questions or suggest ideas to improve/revise student arguments. All skills, from cognitive to affective to psychomotoric, can be improved by efficient and constant practice. In classrooms where expectations of high levels of rigor are standard, students develop proper mathematics reasoning and are keen to identify others’ reasoning and critique it.